In Figure REF , relations between halo \(j_{\Delta }\)  and \(V_{\Delta }\)  are shown.There
is a drawback to a plot of \(j\)  versus \(V\) , namely
both axes incorporate factors of \(V\) , and a relation is expected by construction [1]}.
Because the local relation between galaxy \(V\)  and stellar mass is
tight [2]}, [3]}, there is a similarly tight relation between between \(j\)  and stellar mass [4]},
which is not expected by construction.
Halo \(j\)  is measured within \(R_{200}, R_{2000}\) , and \(R_{20,000}\) , and halo \(V\)  is measured
at \(R_{200}\)  and \(R_{20,000}\) . We do not show results for halo \(V\)  measured at \(R_{2000}\) 
since they do not differ significantly from those for \(R_{200}\)  or \(R_{20,000}\) .
The radii \(R_{2000}\)  and \(R_{20,000}\)  correspond to 34%
and 9% of \(R_{200}\) , respectively, on average Only haloes with more than 100 particles are retained,
except for measurements of \(j_{20,000}\)  and \(V_{20,000}\)  for which haloes with more than
50 particles are used. For these 50-particle haloes, the intrinsic relations remain the same,
but the scatter is increased slightly due to increased Poisson noise.
The shapes of all the distributions are similar in terms
of slope and scatter, and are therefore approximately independent of the radius for which \(j\)  or \(V\)  are measured.
The slope flattens slightly with increasing \(\Delta \) , and the scatter remains about the same.
We will quantify this in the following section.
However, the normalisation is strongly dependent on \(\Delta \) : it decreases
by factors of \(\sim 3\)  and \(\sim 6\)  for 10 and 100-fold increases in \(\Delta \) , respectively.
A decreasing normalisation with increasing \(\Delta \)  is a consequence of how angular momentum is distributed in galactic haloes,
with most of the angular momentum located in the outer parts. As we increase \(\Delta \) , we exclude more and more
of the outer parts of the haloes, and the angular momenta decrease, as illustrated by
the simple analytic treatment in [4]}. In this paper, we quantify this
decrease more precisely using numerical simulations.
